I had the same issues getting .json and font files when deploying a Flutter web app to an Azure App Service. This is what I needed to do:
When ready to deploy, create the deployable web project with:
flutter build web
That creates a folder at the following path:
/project root/build/web
Once that is done - I had to place a web.config file at the root of the web project (same path as above) with these contents. I was having trouble with both .json and font files getting blocked.
<?xml version="1.0"?>
<configuration>
<system.webServer>
<staticContent>
<mimeMap fileExtension=".json" mimeType="application/json" />
<mimeMap fileExtension="woff" mimeType="application/font-woff" />
<mimeMap fileExtension="woff2" mimeType="application/font-woff2" />
<mimeMap fileExtension=".ttf" mimeType="application/octet-stream" />
<mimeMap fileExtension=".ttc" mimeType="application/octet-stream" />
<mimeMap fileExtension=".otf" mimeType="application/octet-stream" />
</staticContent>
</system.webServer>
</configuration>
I'm guessing there could be other static files that could be added to this web.config file as you include them in your project. If so, you'll have to research how those should look in a standard web.config file.